Power variations and limit theorems for stochastic processes controlled by fractional Brownian motions
type: Item
instance of: scholarly article
title: Power variations and limit theorems for stochastic processes controlled by fractional Brownian motions
zbMATH Open document ID: 1548.60058
author: Yanghui Liu
author: Xiaohua Wang
published in: Electronic Journal of Probability
publication date: 2024-10-07T00:00:00Z
review text: In the setting of rough paths, let \((y_t)_{t\in[0,1]}\) be a stochastic process controlled by a fractional Brownian motion with Hurst parameter \(H\leq1/2\). The corresponding power variation of order \(p>0\) is\N\[\N\sum_{k=0}^{n-1}|y_{t_{k+1}}-y_{t_k}|^p\,,\N\]\Nwhere \(0=t_0<t_1<\cdots<t_n=1\). The authors establish limit theorems for these power variations, distinguishing the three cases \(H<1/4\), \(H=1/4\) and \(H>1/4\). The limit theorems are based on a decomposition of the power variation into weighted random sums and remainder terms. When \(H<1/4\) the authors show convergence in probability of the centred power variation at rate \(n^{-2H}\) as \(n\to\infty\). When \(H\geq1/4\) there is convergence in law at rate \(n^{-1/2}\).
reviewed by: Fraser Daly
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G22
zbMATH DE Number: 7922283
zbMATH Keywords: controlled rough path
zbMATH Keywords: discrete rough integral
zbMATH Keywords: estimation of volatility
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: limit theorems
zbMATH Keywords: power variation
MaRDI profile type: Publication
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DOI: 10.1214/24-EJP1179
description: scientific article; zbMATH DE number 7922283